Derivatives (differentiation): the complete guide.
From Class 11 basics to engineering maths. Every derivatives topic, explained the way Vipul Sir teaches it in class: the idea first, then worked examples, then exam practice.
A derivative tells you how fast one quantity changes when another changes. For a function \(y = f(x)\), the derivative, written \(\dfrac{dy}{dx}\) or \(f'(x)\), is the rate of change of \(y\) with respect to \(x\). On a graph, it is the slope of the tangent to the curve at a point. Finding a derivative is called differentiation.
By Vipul Sir · Updated
Pick your level.
Derivatives are taught at every level from Class 11 to university. Choose yours to see which lessons matter for your exam.
Class 11
HSC · CBSE · ISCWhat a derivative means, first principles, and the basic rules (sum, product and quotient) for algebraic and trigonometric functions.
Start with the basics → STEP 02Class 12
HSC · CBSE · ISC · IBChain rule, inverse trigonometric, exponential and log functions, implicit and parametric differentiation, second derivatives, then applications.
Start with the chain rule → STEP 03Diploma
Applied Maths · MSBTEStandard derivatives and rules, then applications: tangents and normals, maxima and minima, and radius of curvature.
Go to applications → STEP 04Engineering & BSc
Engineering Maths · BScSuccessive differentiation, Leibniz's theorem, partial differentiation, Euler's theorem, Jacobians, Taylor series and more.
Go to advanced topics →What is a derivative?
Before any formula, understand what a derivative actually measures. Everything else in this topic builds on this one idea.
Think about a car's speedometer. The distance travelled changes with time, and the speedometer shows how fast it is changing at that exact instant. That instantaneous rate of change is a derivative: speed is the derivative of distance with respect to time.
On a graph, the derivative at a point \(P\) is the slope of the tangent there. To find it, take a second point \(Q\) a small distance \(h\) further along, find the slope of the line \(PQ\), and let \(Q\) slide towards \(P\):
\[f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}\]
This is the derivative from first principles. For example, for \(f(x) = x^2\):
\[f'(x) = \lim_{h\to 0}\frac{(x+h)^2 - x^2}{h} = \lim_{h\to 0}\frac{2xh + h^2}{h} = \lim_{h\to 0}(2x + h) = 2x\]
The derivative of \(y = f(x)\) can be written as \(\dfrac{dy}{dx}\), \(f'(x)\), \(y'\) or \(y_1\). Engineering textbooks often use \(y_1, y_2, \dots, y_n\) for the first, second and \(n\)th derivatives.
Key derivative formulas.
The standard results you should know without thinking. Every method and application in this topic uses them.
| Function | Derivative |
|---|---|
| \(c\) (a constant) | \(0\) |
| \(x^n\) | \(n\,x^{n-1}\) |
| \(\sqrt{x}\) | \(\dfrac{1}{2\sqrt{x}}\) |
| \(\dfrac{1}{x}\) | \(-\dfrac{1}{x^2}\) |
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x\log a\) |
| \(\log x\) | \(\dfrac{1}{x}\) |
| \(\log_a x\) | \(\dfrac{1}{x\log a}\) |
| \(\sin^{-1}x\) | \(\dfrac{1}{\sqrt{1-x^2}}\) |
| Function | Derivative |
|---|---|
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
| \(\tan x\) | \(\sec^2 x\) |
| \(\cot x\) | \(-\operatorname{cosec}^2 x\) |
| \(\sec x\) | \(\sec x\tan x\) |
| \(\operatorname{cosec} x\) | \(-\operatorname{cosec} x\cot x\) |
| \(\cos^{-1}x\) | \(-\dfrac{1}{\sqrt{1-x^2}}\) |
| \(\tan^{-1}x\) | \(\dfrac{1}{1+x^2}\) |
| \(\cot^{-1}x\) | \(-\dfrac{1}{1+x^2}\) |
| Rule | Formula | Lesson |
|---|---|---|
| Sum and difference | \((u \pm v)' = u' \pm v'\) | Coming soon |
| Constant multiple | \((c\,u)' = c\,u'\) | Coming soon |
| Product rule | \((uv)' = u\,v' + v\,u'\) | Coming soon |
| Quotient rule | \(\left(\dfrac{u}{v}\right)' = \dfrac{v\,u' - u\,v'}{v^2}\) | Coming soon |
| Chain rule | \(\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}\) | Read lesson → |
Here \(\log x\) means the natural logarithm (base \(e\)), as in Indian textbooks, and \(u\), \(v\) are functions of \(x\).
Basics.
Start here if derivatives are new to you, or if the rules feel like magic. The tags show which courses each lesson is for.
- 01What is a derivative?Coming soon
Slope, rate of change and the idea behind dy/dx.
- 02Derivative from first principlesComing soon
The limit definition, with solved board-exam examples.
- 03Derivative formulasComing soon
Every standard derivative on one printable sheet.
- 04Continuity and differentiabilityComing soon
When a function has a derivative, and when it doesn't.
Methods.
The techniques you'll use in almost every differentiation question, from the product rule to second derivatives.
- 05Product and quotient ruleComing soon
Differentiate functions that are multiplied or divided.
- 06Chain ruleRead lesson →
Differentiate a function inside a function, with 8 solved examples.
- 07Derivatives of trigonometric functionsComing soon
sin, cos, tan and the rest, and why they work.
- 08Derivatives of inverse trigonometric functionsComing soon
Standard results and the substitution tricks examiners love.
- 09Derivatives of exponential and log functionsComing soon
eˣ, aˣ, log x and their chain rule forms.
- 10Logarithmic differentiationComing soon
For xˣ-type functions and long products.
- 11Implicit differentiationComing soon
Find dy/dx when y isn't written on its own.
- 12Parametric differentiationComing soon
When x and y both depend on a third variable.
- 13Higher-order derivativesComing soon
Second derivatives and "show that" questions.
Applications.
Where derivatives earn their marks: rates of change, tangents, increasing and decreasing functions, and maxima and minima.
- 14Applications of derivatives: overviewComing soon
Every application at a glance, and which exams ask it.
- 15Rate of changeComing soon
Related rates: spheres, ladders, shadows and more.
- 16Tangents and normalsComing soon
Equations of the tangent and normal to a curve.
- 17Increasing and decreasing functionsComing soon
Use the sign of f′(x) to see where a function rises or falls.
- 18Maxima and minimaComing soon
First and second derivative tests, with word problems.
- 19ApproximationsComing soon
Estimate values using differentials.
- 20Radius of curvatureComing soon
How sharply a curve bends at a point.
- 21Rolle's theoremRead lesson →
Statement, conditions and verification questions. The mean value theorems follow in the Calculus course.
Engineering & BSc.
University-level differentiation for Engineering Mathematics and BSc students. Several of these are shared with the Engineering Calculus course.
- 22Successive differentiationComing soon
The nth derivative of standard functions.
- 23Leibniz's theoremComing soon
The nth derivative of a product, step by step.
- 24Partial differentiationComing soon
Derivatives of functions of two or more variables.
- 25Euler's theorem on homogeneous functionsComing soon
Statement, corollaries and solved university questions.
- 26JacobiansComing soon
Definition, properties and change of variables.
- 27Maxima and minima of two variablesComing soon
Stationary points and Lagrange's method of multipliers.
- 28Taylor's and Maclaurin's seriesComing soon
Expand functions as power series.
- 29Indeterminate forms and L'Hôpital's ruleComing soon
Evaluate 0/0, ∞/∞ and the other tricky limits.
Exam practice.
Solved questions in the style of your board or university paper, to use once you've worked through the lessons.
- 30HSC Class 12: important questionsComing soon
Maharashtra Board differentiation and applications, fully solved.
- 31CBSE Class 12: important questionsComing soon
Continuity, differentiability and applications, solved.
- 32Diploma Applied Maths: solved questionsComing soon
MSBTE-style derivative questions with full working.
- 33Mumbai University Engineering Maths: solved questionsComing soon
Past-paper differentiation questions, solved step by step.
Derivatives FAQs.
Quick answers to the questions students ask most often in class.
Is a derivative the same as differentiation?
Almost. The derivative is the result, the rate of change. Differentiation is the process of finding it.
What does dy/dx mean?
It means "the rate of change of \(y\) with respect to \(x\)": how much \(y\) changes for a very small change in \(x\). It is read as "d y by d x".
Why is the derivative of a constant zero?
A constant never changes, so its rate of change is zero. On a graph, \(y = c\) is a horizontal line, and a horizontal line has slope 0.
What is the fastest way to get good at differentiation?
Know the standard formulas without thinking, master the chain rule because it appears everywhere, and practise mixed questions every day rather than one type at a time. Simplify before you differentiate whenever you can.
Where are derivatives used after school?
Throughout engineering maths (partial derivatives, series expansions, differential equations), and in physics (velocity and acceleration), economics (marginal cost) and machine learning (gradients).
Vipul Sir has taught mathematics for over 15 years to HSC, CBSE, ICSE, IGCSE/IB, Diploma, Engineering and BSc students at VVS Classes in Goregaon and Vile Parle, Mumbai. He teaches every class himself, concepts first and exam technique second.
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